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    Mathematical Example of Resource Allocation Optimization

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    If fibers had a higher marginal profit per unit input than gasoline, gallons of crude should be switched from gasoline production to fiber production. Here is concrete example. Suppose the functions are:

    Gasoline: G = 72MG - 1.5MG^2
    Fiber: F = 80MG - 2MG^2

    Here gasoline output is measured in thousands of gallons, fiber output in thousands of square feet, and crude oil in thousands of barrels. The products profits per unit output are $0.50 per gallon for gasoline and $0.75 per square foot for fiber. Then the respective marginal profits are:

    MpG = ($0.50)(72 - 3MG) = 36 - 1.5MG
    MpF = ($0.75)(80 - 4MF) = 60 - 3MF

    Setting these equal to each other and rearranging gives:

    MF = 0.5MG + 8

    Solving this equation and the constraint MG + MF = 20 implies MG = 8 thousand barrels and MF = 12 thousand barrels. This allocation generates 480 thousand gallons of gasoline and 672 thousand square feet of fiber. The firm's total profit is $744 (less the cost of the crude).

    Question: Find the optimal crude oil allocation in the preceding example if the profit associated with fiber were cut in half, that is, fell to $0.375 per square foot.

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    Solution Preview

    Set up the equations like in the example problem, but with the difference that the marginal profit of fiber is $0.375 instead of $0.75:

    MpG = ($0.50)(72-3MG) = 36 - 1.5MG
    MpF = ($0.375)(80-4MF) = 30 - 1.5MF

    Setting them equal to ...

    Solution Summary

    Crude oil can be used for gasoline production or fiber production. Given production functions and profit margins for gasoline and fibers, calculate the optimum allocation of crude oil to maximize profits for a firm that produces both products.